Clustering problems such as $k$-Median, and $k$-Means, are motivated from applications such as location planning, unsupervised learning among others. In such applications, it is important to find the clustering of points that is not ``skewed'' in terms of the number of points, i.e., no cluster should contain too many points. This is modeled by capacity constraints on the sizes of clusters. In an orthogonal direction, another important consideration in clustering is how to handle the presence of outliers in the data. Indeed, these clustering problems have been generalized in the literature to separately handle capacity constraints and outliers. To the best of our knowledge, there has been very little work on studying the approximability of clustering problems that can simultaneously handle both capacities and outliers. We initiate the study of the Capacitated $k$-Median with Outliers (C$k$MO) problem. Here, we want to cluster all except $m$ outlier points into at most $k$ clusters, such that (i) the clusters respect the capacity constraints, and (ii) the cost of clustering, defined as the sum of distances of each non-outlier point to its assigned cluster-center, is minimized. We design the first constant-factor approximation algorithms for C$k$MO. In particular, our algorithm returns a (3+\epsilon)-approximation for C$k$MO in general metric spaces, and a (1+\epsilon)-approximation in Euclidean spaces of constant dimension, that runs in time in time $f(k, m, \epsilon) \cdot |I_m|^{O(1)}$, where $|I_m|$ denotes the input size. We can also extend these results to a broader class of problems, including Capacitated k-Means/k-Facility Location with Outliers, and Size-Balanced Fair Clustering problems with Outliers. For each of these problems, we obtain an approximation ratio that matches the best known guarantee of the corresponding outlier-free problem.

翻译：聚类问题如$k$-中位数和$k$-均值受位置规划、无监督学习等应用驱动。在这些应用中，找到点数上不"偏斜"的聚类至关重要，即每个聚类不应包含过多数据点。这通过聚类大小的容量约束进行建模。在正交方向上，聚类中另一个重要考虑因素是如何处理数据中的离群值。事实上，现有文献已分别对容量约束和离群值处理对聚类问题进行泛化推广。据我们所知，同时处理容量约束和离群值的聚类问题的近似性研究极少。我们首次研究带离群值的容量限制$k$-中位数问题（C$k$MO）。该问题要求将除$m$个离群点外的所有数据点聚成最多$k$个聚类，使得：（i）聚类满足容量约束；（ii）聚类成本（定义为每个非离群点到其分配聚类中心距离之和）最小化。我们为C$k$MO设计了首个常数因子近似算法。特别地，我们的算法在一般度量空间中获得$(3+\epsilon)$-近似比，在常维欧几里得空间中获得$(1+\epsilon)$-近似比，运行时间为$f(k, m, \epsilon) \cdot |I_m|^{O(1)}$，其中$|I_m|$表示输入规模。我们还将这些结果推广到更广泛的问题类，包括带离群值的容量限制$k$-均值/$k$-设施选址问题，以及带离群值的规模平衡公平聚类问题。对于这些问题的每一个，我们获得的近似比均匹配对应无离群值问题已知的最佳保证。